Page 428 - Advanced Linear Algebra
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412 Advanced Linear Algebra
Let us pause briefly to consider an important application of strictly positive
solutions to a system (% ~ . If ? ~ ²% ÁÃÁ% ³ is a strictly positive solution
to this system, then so is the vector
& ~ ² % Á Ã Á % ³ ~ ² ~ Á Ã Á ? ³
' '% %
which is a probability distribution , that is, and ~ . Moreover,
if is a strongly positive solution, then has the property that each probability
?
&
is positive.
Now, the product ( is the expected value of the columns of with respect to
(&
the probability distribution . Hence, (% ~ has a strictly (strongly) positive
&
solution if and only if there is a strictly (strongly) positive probability
distribution for which the columns of have expected value . If each column
(
of represents the possible payoffs from a game of chance, where each row is a
(
different possible outcome of the game, then the game is fair when the expected
value of the columns is . Thus, ( % ~ has a strictly (strongly) positive
solution if and only if the game with payoffs and probabilities is fair.
?
(
?
As another (related) example, in discrete option pricing models of mathematical
finance, the absence of arbitrage opportunities in the model is equivalent to the
fact that a certain vector describing the gains in a portfolio does not intersect the
strictly positive orthant in s . As we will see in this chapter, this is equivalent
to the existence of a strongly positive solution to a homogeneous system of
equations. This solution, when normalized to a probability distribution, is called
a martingale measure .
Of course, the equation (% ~ has a strictly positive solution if and only if
ker²(³ contains a strictly positive vector, that is, if and only if
ker²(³ ~ RowSpace ²(³
meets the strictly positive orthant in s . Thus, we wish to characterize the
subspaces of s : for which : meets the strictly positive orthant in s , in
symbols,
:q s £ J
b
(
for these are precisely the row spaces of the matrices for which % ( ~ has a
strictly positive solution. A similar statement holds for strongly positive
solutions.
Looking at the real plane s , we can divine the answer with a picture. A one-
dimensional subspace of s has the property that its orthogonal complement
:
: meets the strictly positive orthant quadrant in s
%
: ( ) if and only if is the -
axis, the -axis or a line with negative slope. For the case of the strongly
&
